Solution - Solving quadratic inequalities using the quadratic formula

To find the roots of a quadratic equation, plug its coefficients ($$a$$, $$b$$ and $$c$$ ) into the quadratic formula:

$$x=\left(-b\pm sqrt\left(b^2-4ac\right)\right)/\left(2a\right)$$

$$x=\left(-1*-9\pm sqrt\left(-9^2-4*1*-14\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(81-4*1*-14\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(81-4*-14\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(81--56\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(81+56\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(137\right)\right)/\left(2*1\right)$$

$$x=\left(-1*-9\pm sqrt\left(137\right)\right)/\left(2\right)$$

$$x=\left(9\pm sqrt\left(137\right)\right)/2$$

to get the result:

$$x=\left(9\pm sqrt\left(137\right)\right)/2$$

Simplify $$137$$ by finding its prime factors:

The prime factorization of $$137$$ is $$137$$

$$x=\left(9\pm sqrt\left(137\right)\right)/2$$

The ± means two roots are possible.

Separate the equations:
$$x_1=\left(9+sqrt\left(137\right)\right)/2$$ and $$x_2=\left(9-sqrt\left(137\right)\right)/2$$

$$x_1=\left(9+sqrt\left(137\right)\right)/2$$

$$x_1=\left(9+sqrt\left(137\right)\right)/2$$

$$x_1=\left(9+11.705\right)/2$$

$$x_1=\left(9+11.705\right)/2$$

$$x_1=\left(20.705\right)/2$$

$$x_1=\frac{20.705}{2}$$

$$x_1=10.352$$

$$x_2=\left(9-sqrt\left(137\right)\right)/2$$

$$x_2=\left(9-11.705\right)/2$$

$$x_2=\left(9-11.705\right)/2$$

$$x_2=\left(-2.705\right)/2$$

$$x_2=-\frac{2.705}{2}$$

$$x_2=-1.352$$

To find the intervals of a quadratic inequality, we start by finding its parabola.

The roots of the parabola (where it meets the x-axis) are: -1.352, 10.352.

Since the $$a$$ coefficient is positive ($$a$$=1), this is a "positive" quadratic inequality and the parabola points upward, like a smile!

If the inequality sign is ≤ or ≥ , then the intervals include the roots and we use a solid line. If the inequality sign is < or > the intervals do not include the roots and we use a dotted line.

Since $$x^2-9x-14>0$$ has a $$>$$ inequality sign, we look for the parabola intervals that are above the x-axis.

Solution: $$$$

Interval notation: $$$$